Why do we need the limit to exist for the slope of the tangent line?

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SUMMARY

The discussion centers on the necessity of the existence of limits for the slope of the tangent line in calculus. Specifically, the limit is defined as m = lim (x → a) (f(x) - f(a)) / (x - a). Participants highlight that both the left-hand limit and right-hand limit must exist and be equal for a tangent line to be defined. An example provided is the function f(x) = |x| at x = 0, where the left-hand and right-hand limits differ, illustrating that the tangent line cannot be established in such cases.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with secant lines and their relationship to tangent lines
  • Basic knowledge of piecewise functions, specifically f(x) = |x|
  • Concept of left-hand and right-hand limits
NEXT STEPS
  • Study the definition and properties of limits in calculus
  • Explore the concept of continuity and its relation to tangent lines
  • Investigate piecewise functions and their differentiability
  • Learn about the formal definition of the derivative and its applications
USEFUL FOR

Students studying calculus, educators teaching limit concepts, and anyone seeking to understand the foundational principles of tangent lines and differentiability.

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Homework Statement



My textbook says that the slope of the tangent line at a point can be expressed as a limit of secant lines:

[tex] m = \underset{x \rightarrow a}{\lim} \, \frac{f(x) - f(a)}{x - a} \, .[/tex]

If x > a and we approach a from the right, why do we have to insist that this limit exists? Why can't we settle for the right-handed limit instead?

Homework Equations

The Attempt at a Solution



I'm really not sure why the left limit needs to exist. Any help is appreciated.
 
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Because sometimes the left limit is different from the right limit. Then the limit doesn't exist, and you don't have anyone tangent line. Look at the function [tex]f(x) = |x|[/tex] at [itex]x = 0[/itex]. What is the left hand tangent line limit? What's the right one?
 

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